Adaptive method of lines

Alain Vande Wouwer, A Vande Wouwer, Ph. Saucez, W.E. Schiesser158488231X, 9781584882312

The general Method of Lines (MOL) procedure provides a flexible format for the solution of all the major classes of partial differential equations (PDEs) and is particularly well suited to evolutionary, nonlinear wave PDEs. Despite its utility, however, there are relatively few texts that explore it at a more advanced level and reflect the method’s current state of development.Written by distinguished researchers in the field, Adaptive Method of Lines reflects the diversity of techniques and applications related to the MOL. Most of its chapters focus on a particular application but also provide a discussion of underlying philosophy and technique. Particular attention is paid to the concept of both temporal and spatial adaptivity in solving time-dependent PDEs. Many important ideas and methods are introduced, including moving grids and grid refinement, static and dynamic gridding, the equidistribution principle and the concept of a monitor function, the minimization of a functional, and the moving finite element method. Applications addressed include shallow water flow, combustion and flame propagation, transport in porous media, gas dynamics, chemical engineering processes, solitary waves, and magnetohydrodynamics.As the first advanced text to represent the modern era of the method of lines, this monograph offers an outstanding opportunity to discover new concepts, learn new techniques, and explore a wide range of applications.

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Table of contents :
Adaptive Method of LINES……Page 1
Preface……Page 3
Contributors……Page 8
Contents……Page 11
1.1 Classification of Partial Differential Equations……Page 17
Contents……Page 0
1.2 The Method of Lines……Page 26
1.2.1 Spatial Discretization……Page 28
1.3 Adaptive Grid Methods……Page 30
1.3.1 Grid Adaptation Criteria……Page 32
1.3.2 Static vs. Dynamic Gridding……Page 33
1.3.4 Grid Regularity……Page 34
1.4.1 Case Study 1……Page 35
1.4.2 Case Study 2……Page 37
1.4.3 Case Study 3……Page 38
1.4.4 Case Study 4……Page 40
1.4.5 Case Study 5……Page 43
1.4.6 Case Study 6……Page 47
1.5 Summary……Page 50
References……Page 51
2.2 Adaptive Grid Refinement……Page 54
2.2.1 Grid Equidistribution with Constraints……Page 55
2.3 Application Examples……Page 57
2.3.1 The Nonlinear Schrödinger Equation……Page 58
2.3.2 The Derivative Nonlinear Schrödinger Equation……Page 66
2.3.3 The Korteweg-de Vries Equation……Page 67
2.3.4 The Korteweg-de Vries-Burgers Equation……Page 71
2.3.5 KdV-Like Equations: The Compactons……Page 73
References……Page 77
3.1 Introduction……Page 80
3.2 Equal-Width Equation……Page 85
3.3 Numerical Solution Procedure……Page 88
3.4.1 Single Solitary Waves……Page 100
3.4.2 Inelastic Interaction of Solitary Waves……Page 110
3.4.3 Gaussian Pulse Breakup into Solitary Waves……Page 117
3.4.4 Formation of an Undular Bore……Page 124
3.5 Concluding Remarks……Page 127
References……Page 129
4.1 Introduction……Page 132
4.2 The Equations of Magnetohydrodynamics……Page 133
4.3.1 The MHD Equations in 1D……Page 134
4.3.2 The Adaptive Grid Method in One Space Dimension……Page 135
4.3.3 Numerical Results……Page 138
4.4.1 2D Magnetic Field Evolution……Page 144
4.4.2 Adaptive Grids in Two Space Dimensions……Page 145
4.5 Conclusions……Page 150
References……Page 151
5.1 Introduction……Page 153
5.2 Two-Step Numerical Modeling……Page 155
5.3 1-D Shallow-Water Equations……Page 159
5.4 Compatible Discretization……Page 162
5.4.1 Discretized Shallow-Water Equations……Page 165
5.4.2 Iterative Solution Algorithm……Page 168
5.5 Error Analysis……Page 170
5.5.1 Error Analysis in Space……Page 171
5.5.3 Error in Discretized Shallow-Water Equations……Page 177
5.6 Error-Minimizing Grid Adaptation……Page 179
5.7.1 Steady-State Application……Page 183
5.7.2 Unsteady Application……Page 189
5.8 Conclusions……Page 191
References……Page 192
6.1 Introduction……Page 195
6.2 Modeling Flow and Transport in Rivers……Page 198
6.3.1 Network Approach……Page 201
6.3.2 Space Discretization……Page 202
6.3.3 Time Integration……Page 203
6.4 Adaptive Space Mesh Strategies……Page 208
6.4.1 Extension to 2D Problems……Page 210
6.5 Applications……Page 213
6.6 Conclusion……Page 215
References……Page 216
7.1.1 Overview: Adaptive Gridding……Page 220
7.1.2 Overview: Free Surface Model……Page 221
7.2 Governing Equations……Page 222
7.2.1 Projection Method……Page 223
7.3 Discretization……Page 224
7.3.1 Thickness of the Interface……Page 226
7.4 Coupled Level Set Volume of Fluid Advection Algorithm……Page 227
7.5.1 Projection Step in General Geometries……Page 229
7.5.2 Contact-Angle Boundary Condition in General Geometries……Page 232
7.6 Adaptive Mesh Refinement……Page 233
7.6.1 Time-Stepping Procedure for Adaptive Mesh Refinement……Page 234
7.7.1 Axisymmetric Jetting Convergence Study……Page 235
7.7.2 3D Ship Waves……Page 237
References……Page 240
8.1 Introduction……Page 245
8.2 Moving Finite Elements……Page 247
8.2.2 Minimization Principles and Weak Forms……Page 249
8.3 A Local Approach to Variational Principles……Page 250
8.3.2 A Local Approach to Best Fits……Page 251
8.3.3 Direct Optimization Using Minimization Principles……Page 252
8.3.4 A Discrete Variational Principle……Page 253
8.4.1 Least-Squares Moving Finite Elements……Page 255
8.4.2 Properties of the LSMFE Method……Page 256
8.4.4 Least-Squares Finite Volumes……Page 257
8.4.5 Example……Page 259
8.5 Conservation Laws by Least Squares……Page 261
8.5.1 Use of Degenerate Triangles……Page 263
8.5.2 Numerical Results for Discontinuous Solutions……Page 264
8.6 Links with Equidistribution……Page 266
8.6.1 Approximate Multidimensional Equidistribution……Page 267
8.6.3 Approximate Equidistribution and Conservation……Page 268
8.7 Summary……Page 270
References……Page 271
9.1 Introduction……Page 274
9.2.1 The Discretization……Page 277
9.2.2 The Adaptive Algorithms……Page 278
9.3.1 The Discretization……Page 284
9.3.2 The Monotone Convergence……Page 286
9.3.3 The Error Control and Stopping Criterion……Page 290
9.4 Computational Examples and Conclusions……Page 292
References……Page 303
10.1 Introduction……Page 306
10.2 Linearly Implicit Methods……Page 307
10.3 Multilevel Finite Elements……Page 312
10.4.1 Stability of Flame Balls……Page 313
10.4.2 Brine Transport in Porous Media……Page 316
10.5 Conclusion……Page 323
References……Page 324
11.1 Introduction……Page 328
11.2 Spatial Discretization and Time Integration……Page 330
11.3 Space-Time Error Balancing Control……Page 332
11.4 Fixed and Adaptive Mesh Solutions……Page 333
11.5 Atmospheric Modeling Problem……Page 335
11.6 Triangular Finite Volume Space Discretization Method……Page 337
11.7 Time Integration……Page 340
11.8 Mesh Generation and Adaptivity……Page 341
11.9 Single-Source Pollution Plume Example……Page 343
11.10 Three Space Dimensional Computations……Page 346
11.11.1 Flux Evaluation Using Edge-Based Operation……Page 347
11.11.2 Adjustments of Wind Field……Page 349
11.11.4 Diffusion Scheme……Page 350
11.12 Mesh Adaptation……Page 351
11.13 Time Integration for 3D Problems……Page 352
11.14 Three-Dimensional Test Examples……Page 353
11.14.1 Grid Adaptation……Page 355
11.15 Discussions and Conclusions……Page 357
References……Page 359
12.1 Introduction……Page 363
12.2.2 Continuum Model Equations……Page 365
12.2.3 Initial and Boundary Conditions……Page 366
12.2.5 Dimensionless Equations……Page 367
12.2.6 Method of Solution……Page 369
12.3.2 Numerical Solutions……Page 371
12.4 Discussion……Page 376
References……Page 378
13.1 Introduction……Page 380
13.2 Architecture of the Simulation Environment Diva……Page 381
13.2.1 The Diva Simulation Kernel……Page 382
13.2.3 Symbolic Preprocessing Tool……Page 388
13.3 MOL Discretization of PDE and IPDE……Page 389
13.3.1 Finite-Difference Schemes……Page 392
13.3.2 Finite-Volume Schemes……Page 393
13.3.3 High-Resolution Schemes……Page 395
13.3.4 Equidistribution Principle Based Moving Grid Method……Page 396
13.4 Symbolic Preprocessing for MOL Discretization……Page 397
13.4.1 Mathematica Data Structure……Page 398
13.4.2 Procedure of the MOL Discretization……Page 400
13.5 Application Examples……Page 401
13.5.1 Circulation-Loop-Reactor Model……Page 402
13.5.2 Moving-Bed Chromatographic Process……Page 406
13.6 Conclusions and Perspectives……Page 411
References……Page 412